I've seen public Trello boards[1]. We use Jira tickets - you can have a customer feedback project that enables customers to create tickets without seeing the full board[2].
I want a seamless service where customers can easily submit and track their issues with our APIs during the testing phase. One of the ways I am thinking of is creating a public repo on GitHub for issue management but that is something client might not appreciate.
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[1] https://blog.trello.com/how-to-transform-trello-into-a-power...
[2] https://medium.com/thinktilt/five-options-for-creating-jira-...